Modified energy functionals and the NLS approximation (Q525517)

The authors consider the relation between 2D water wave problems and nonlinear Schrödinger equations. In 2D water wave problems, one studies the irrotational flow of a homogeneous and inviscid fluid. One considers the following model system: \[ \partial_t u(x,t)=\Omega v, \;\partial_t u(x,t)=\Omega u +\Omega (u^2 ). \] Here \(\Omega\) is defined by \(\widehat{\Omega u}(x,t)=i\mathrm{ sgn}\!\;(k)\sqrt{k\mathrm{ tanh }\!\;k}\hat u (k,t)\) using the Fourier transform (Here \(k\) denote the dual variables of \(x\)). On the other hand, one considers the solution \(A(X,T)\) of the nonlinear Schrödinger equation \[ \partial_T A(X,T)=i\nu_1 \partial^2_X A+i\nu_2 A|A|^2 \] for some constants \(\nu_j =\nu_j (k_0 )\) and a small parameter \(\epsilon\). Considering the Cauchy problems for these two equations, the authors prove that the solution \((u(x,t),v(x,t))\) to the former problem is approximated by \[ \epsilon\Psi_{NLS} =\epsilon A(\epsilon (x+c_0 t),\epsilon^2 t)e^{i(k_0 x+\omega_0 t)}\phi (k_0 ) \] using the solution \(A(X,T)\) to the latter one and a vector \(\phi (k_0 )\in\mathbb{C}^2\). One of the authors studied this problem previously, and the present article improves the former result by using an energy method.